LEM. New Results on Betting Strategies, Market Selection, and the Role of Luck. Giulio Bottazzi Daniele Giachini
|
|
- Dora Edwards
- 6 years ago
- Views:
Transcription
1 LEM WORKING PAPER SERIES New Results on Betting Strategies, Market Selection, and the Role of Luck Giulio Bottazzi Daniele Giachini Institute of Economics, Scuola Sant'Anna, Pisa, Italy 2018/08 February 2018 ISSN(ONLINE)
2 New Results on Betting Strategies, Market Selection, and the Role of Luck Giulio Bottazzi 1 and Daniele Giachini 1 1 Istituto di Economia, Scuola Superiore Sant Anna, Piazza Martiri della Libertà 33, Pisa - Italy February 23, 2018 Abstract We consider a repeated betting market populated by two agents who wage on a binary event according to generic betting strategies. We derive new simple criteria to establish the relative wealth of the two agents in the long run, only based on the odds they believe fair and how much they would bet when the odds are equal to the ones the other agent believes fair. Using our criteria, we show that for a large class of betting strategies it is generically possible that the ultimate winner is only decided by luck. As an example, we apply our conditions to the case of CRRA betting. Keywords: Bounded Rationality, Betting Strategies, Market Selection, Recurrent Processes, Luck. JEL Classification: C60, D53, G11, G12 1 Introduction Consider a repeated market for betting where two agents wage on the outcomes of a binary event. Agent behavior is described by generic betting strategies that depend upon prevailing odds. If the odds are fixed, the strategy that guarantees optimal wealth growth is the Kelly rule (Kelly, 1956; Breiman, 1961). If the odds depend on agents bets via the parimutuel procedure, Beygelzimer et al. (2012) show that in a population of Kelly bettors, the one with the most accurate beliefs accrues all the wealth and asymptotically dominates the market. This is a particular case of the result derived by Blume and Easley (1992) and Blume 1
3 and Easley (2009) in the equivalent setting of inter-temporal general equilibrium models with short lived securities. In a similar framework, Evstigneev et al. (2002) clarify that an agent adopting the Kelly strategy and having correct beliefs will dominate the economy irrespective of the strategies adopted by other agents. The global dominance of the Kelly strategy with perfect information being established, it remains to understand what happens when agents do not bet according to Kelly and/or are not perfectly informed. Kets et al. (2014) consider the case of fractional Kelly and CRRA bettors, providing a few tentative results based on numerical simulations. Bottazzi and Giachini (2016) derive sufficient and, apart from hairline cases, necessary conditions for strategy dominance or survival in a market of fractional Kelly bettors. In the present paper we propose more general criteria that can be applied to generic strategies depending on prevailing market odds. The new criteria are simple and abstract from strategy specific details. It is sufficient to know the odds bettors consider fair and the amount each bettor is willing to bet when the odds are equal to the ones the other bettor believes fair to understand if one bettor will eventually dominate the market or, conversely, if the two bettors will asymptotically retain a finite, and fluctuating, amount of wealth. Interestingly, by considering generic strategies, one recovers the traditional role of luck in the game of chance, largely neglected in the previously mentioned studies. Indeed, in our general setting, it is generically possible that the ultimate fate of a bettor is not only decided by the adopted strategy, but also by the specific realized sequence of binary events. As an example, we apply the new criteria to the case of CRRA bettors. 2 Model Consider two agents who make a sequence of bets against each other on binary events. The outcome of the event s t {0,1} is an independent Bernoulli trial with success probability π : s t = 1 means that the event occurs while s t = 0 that it does not. In each round, agent i {1,2} has to choose the fraction of wealth to be wagered b i t and the side of the bet σ i t {0,1}, where 1 means betting on the occurrence of the event while 0 betting against it. We assume the amount bet is redistributed among the winners according to the parimutuel procedure, that is, proportionally to how much they have bet, without any house-take. Let p t be the prevailing inverse odd ratio at round t for the occurrence of the event. Thus if s t = 1 the agent betting on the occurrence of the event receives 1/p t times the amount bet while if s t = 0 the agent betting against the occurrence of the event receives 1/(1 p t ) times the amount bet. Agents betting strategies are based on prevailing odds and they try to maximize their gain by increasing their bet when they perceive favorable opportunities. Following Kets et al. (2014), we assume 2
4 that for each agent i there exists a fair inverse odd p i (0,1) and a continuous function b i [0,1) such that σ i t = 1 if p t < p i, σ i t = 0 if p t > p i, b i ( p i ) = 0 and b i (p t ) > 0 when p t p i. 1 Without loss of generality we set p 1 < p 2. Thus, if w i t 1 is the wealth of agent i {1,2} before the event at time t is realized, the prevailing inverse odd p t is set by the equation w 1 t 1b 1 (p t ) = w 2 t 1b 2 (p t ) (1) being always σt 1 = 0 and σt 2 = 1. We require that the functions b i are such that (1) admits one and only one solution. This is for instance the case if they are monotonic, strictly concave or strictly convex on the set of attainable prices. The amount of wealth that is not bet is invested in a risk-less asset that pays no interest. Hence, after the event at round t is realized, the wealth of agents is updated according to ( wt i = (1 b i (p t ))wt 1 i +δ st,i 1wt 1b i i δi,1 (p t ) + δ ) i,2 (2) 1 p t p t where δ a,b is the Kronecker delta. Since the house takes no fee, the aggregate wealth is constant and we set w t = w 1 t +w 2 t = 1 such that p t [ p 1, p 2 ] and p t = p i if and only if w i t = 1. 3 Long-run Selection The dynamics of wealth described by (2) can lead to two different outcomes: either a single agent accrues all the wealth and dominates the market or both agents indefinitely survive, each with a positive, and fluctuating, fraction of wealth. In general, the fate of an agent could depend on the specific sequence of realizations of the random variable s t. Let σ = {s 1,s s,...} denote a realization of the Bernoulli processandletw i t(σ)betheassociatedsequenceofagenti swealth. Thelong-term outcomes of the repeated betting are formalized in the following. Definition 3.1. Agent i (asymptotically) dominates on σ if lim t w i t(σ) = 1. Agent i (asymptotically) survives on σ if limsup t w i t(σ) > 0. Agenti(asymptotically)dominates iflim t w i t(σ) = 1foralmostallσ. Agent i (asymptotically) survives if limsup t w i t > 0 for almost all σ. Notice that dominance implies survival. If one agent dominates, the other cannot survive and we say that it vanishes. At the same time, if one agent survives the other cannot dominate. 1 We rule out the possibility that agents bet all their wealth as this would lead them to wealth zero almost surely 3
5 We will show that in order to decide survival or dominance of agents, it is not generically necessary to know all the details of the investment strategies, but simply the Bernoulli probability π, the inverse odds considered fair by the two agents, p i, and two positive numbers, b 1 ( p 2 ) and b 2 ( p 1 ), representing the fraction of wealth one agent bets if the odds are equal to those the other agent would consider fair. Proposition 3.1. Consider the quantities and µ 1 = π log p1 +(1 p 1 )b 2 ( p 1 ) p 1 +(1 π )log(1 b 2 ( p 1 )) (3) µ 2 = π log(1 b 1 ( p 2 )) (1 π )log 1 p2 + p 2 b 1 ( p 2 ) 1 p 2. (4) If agents betting strategies satisfy the requirements of Section 2, then i) if µ 1 > 0 and µ 2 > 0 agent 2 dominates and lim t p t = p 2 almost surely; ii) if µ 1 < 0 and µ 2 < 0 agent 1 dominates and lim t p t = p 1 almost surely; iii) if µ 2 < 0 and µ 1 > 0 both agents survive; iv) if µ 2 > 0 and µ 1 < 0 either agent 1 dominates or agent 2 dominates depending on the realization of the Bernoulli process. Proof. Considertheprocess{z t = log(w 2 t/w 1 t)}andnoticethatµ 1 = lim z E[z t+1 z t z t = z] and µ 2 = lim z + E[z t+1 z t z t = z]. Define the (conditional) increment g(p,s) = z t+1 z t when p t = p and s t+1 = s. From (2) remembering that, by hypothesis, b 1 (p) and b 2 (p) cannot be both zero for the same p and are continuous, it is immediate to see that log 1 B 2 1+B 1 p 2 /(1 p 2 ) < g(p,0) < 0 < g(p,1) < log 1+B2 (1 p 1 )/ p 1 1 B 1, whereb i = max{b i (x) p 1 x p 2 }. Thustheincrementsg arefiniteandbounded and Theorems 2.2, 3.1 and 3.2 of Bottazzi and Dindo (2015) can be applied to the process {z t }. If µ 1 > 0 and µ 2 > 0 then lim t z t = +, whence i). If µ 1 < 0 and µ 2 < 0 then lim t z t =, whence ii). If µ 2 < 0 and µ 1 > 0 then there exists a finite interval A such that z t A almost surely for any t, whence iii). If µ 2 > 0 and µ 1 < 0 then on any Bernoulli sequence either lim t z t = + or lim t z t =, whence iv) 4
6 Figure 1: Agents betting strategies. In both panels we set p 1 = 0.3 and p 2 = 0.75, in the first plot we have γ 1 = γ 2 = 2 while in the second it is γ 1 = γ 2 = 0.5. It is immediate to see that if π > p 2 we are in case i) while if π < p 1 we are in case ii), recovering a result in Kets et al. (2014). 2 Notice that the dominance of one agent in case iv) is not (only) realized on specific zero-measure sequences, like the sequence of 1 s or the sequence of 0 s, but on sets of sequences with finite probability. This is where the luck enters into the picture: both agents might dominate, but only the blind goodness will decide who. 4 Example with CRRA Bettors The betting strategies introduced in Section 2 are flexible enough to accommodate several behavioral prescriptions. As an illustrative example, we consider the case in which agents bet to maximize the expected utility of wealth using a power utility function with Constant Relative Risk Aversion (CRRA). Call γ i > 0 the relative risk aversion coefficient of agent i and π i the subjective probability (belief) that agent i assigns to the realization of the event, which is precisely the inverse odd that agent i would consider fair. Assuming π 1 < π 2, for p t [π 1,π 2 ] agent 1 bets against the occurrence of the event a fraction of wealth b 1 which maximizes π 1 (1 b 1 ) 1 γ1 +(1 π 1 )(1 b 1 p t /(1 p t )) 1 γ1 to obtain b 1 (p t ) = (p t (1 π 1 )) 1 γ 1 (π 1 (1 p t )) 1 γ 1 (p t (1 π 1 )) 1 γ 1 +p t (π 1 ) 1 γ 1 (1 p t ) 1 γ1 γ 1. (5) 2 The definitions of survival and dominance in Kets et al. (2014) are weaker than the ones adopted here. Given the relative simplicity of the considered process, however, their conclusions are still valid under Definition
7 a b c d Figure 2: Dominance, survival and vanishing for different combinations of fair inverse odd ratios p i. 1D: agent 1 dominates; 2D: agent 2 dominates; 1,2S: both agents survive; 1D2D: either agent 1 or agent 2 dominates. Panel a: π = 0.45, γ 1 = γ 2 = 2; panel b: π = 0.45, γ 1 = γ 2 = 0.5; panel c: π = 0.45, γ 1 = 2, γ 2 = 0.5; panel d: π = 0.75, γ 1 = 0.5, γ 2 = 2. 6
8 Conversely agent 2 bets in favor of the realization of the event a fraction of wealth b 2 which maximizes π 2 (1+b 2 (1 p t )/p t ) 1 γ2 +(1 π 2 )(1 b 2 ) 1 γ2 to obtain b 2 (p t ) = (π 2 (1 p t )) 1 γ 2 (p t (1 π 2 ) 1 γ 2 (π 2 (1 p t )) 1 γ 2 +(1 p t )(1 π 2 ) 1 γ 2 (p t ) 1 γ2 γ 2. (6) The positive risk aversion implies that agents never bet the totality of their wealth. Figure 1 provides two examples of how agents betting strategies vary depending on the inverse odd ratio. In the effective price support, betting strategies are always continuous and strictly concave. Figure 2 reports the long-run selection outcomes inferred using the conditions from Proposition 3.1. Depending on agents risk aversion and beliefs any case of Proposition 3.1 may generically occur. Notice how low risk aversion and asymmetric beliefs enhance the role of luck in deciding the ultimate winner. 5 Conclusion In this paper we consider a market for bets where two agents repeatedly wage on an uncertain event with two possible outcomes using generic betting strategies. We propose simple criteria to decide about the asymptotic amount of wealth of the two bettors, based on a few quantities: their fair odds and the amount they are willing to bet at the fair odds of the opponent. When generic betting strategies are considered, three outcomes are possible in the long-run: 1) one bettor accrues all the wealth with probability 1; 2) both bettors survive with a positive and fluctuating amount of wealth or 3) one of the two bettor eventually accrues all the wealth with finite probability. In the third case, luck recovers the role of ultimate arbiter, traditionally attributed to it in games of chance. Notice that if one confines the analysis to specific families of strategies, like Kelly or fractional Kelly strategies, the third outcome becomes non-generic or disappear(bottazzi and Giachini, 2016). This explains why it was largely unobserved or not satisfactorily discussed in previous studies. References Beygelzimer, A., J. Langford, and D. M. Pennock (2012). Learning performance of prediction markets with kelly bettors. In Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems-Volume 3, pp International Foundation for Autonomous Agents and Multiagent Systems. 7
9 Blume, L. and D. Easley (1992). Evolution and market behavior. Journal of Economic Theory 58, Blume, L. and D. Easley (2009). The market organism: Long-run survival in markets with heterogenous traders. Journal of Economic Dynamics and Control 33, Bottazzi, G. and P. Dindo(2015). Drift criteria for persistence of discrete stochastic processes on the line. LEM Papers Series 2015/26, Laboratory of Economics and Management (LEM), Sant Anna School of Advanced Studies, Pisa, Italy. Bottazzi, G. and D. Giachini (2016). Far from the madding crowd: Collective wisdom in prediction markets. LEM Papers Series 2016/14, Laboratory of Economics and Management (LEM), Sant Anna School of Advanced Studies, Pisa, Italy. Breiman, L. (1961). Optimal gambling systems for favorable games. Proceedings of the 4th Berkley Symposium on Mathematical Statistics and Probability 1, Evstigneev, I., T. Hens, and K. Schenk-Hoppé (2002). Market selection of financial trading strategies: Global stability. Mathematical Finance 12, Kelly, J. (1956). A new interpretation of information rates. Bell System Technical Journal 35, Kets, W., D. M. Pennock, R. Sethi, and N. Shah (2014). Betting strategies, market selection, and the wisdom of crowds. In Twenty-Eighth AAAI Conference on Artificial Intelligence. 8
Rationality and Asset Prices under Belief Heterogeneity
Rationality and Asset Prices under Belief Heterogeneity Daniele Giachini Istituto di Economia, Scuola Superiore Sant Anna Piazza Martiri della Libertà 33, 56127 Pisa - Italy March 9, 2018 Abstract In this
More informationWealth Selection in a Financial Market with Heterogeneous Agents
Wealth Selection in a Financial Market with Heterogeneous Agents Mikhail Anufriev a,b, Pietro Dindo b,a, a CeNDEF, Department of Quantitative Economics, University of Amsterdam, Roetersstraat, NL-08 WB
More informationA New Interpretation of Information Rate
A New Interpretation of Information Rate reproduced with permission of AT&T By J. L. Kelly, jr. (Manuscript received March 2, 956) If the input symbols to a communication channel represent the outcomes
More informationPrice and Wealth Dynamics in a Speculative Market with an Arbitrary Number of Generic Technical Traders
Price and Wealth Dynamics in a Speculative Market with an Arbitrary Number of Generic Technical Traders Mikhail Anufriev Giulio Bottazzi S.Anna School for Advanced Studies, Pisa, Italy February 20, 2005
More informationPreliminary Results on Social Learning with Partial Observations
Preliminary Results on Social Learning with Partial Observations Ilan Lobel, Daron Acemoglu, Munther Dahleh and Asuman Ozdaglar ABSTRACT We study a model of social learning with partial observations from
More information1. Introduction. 2. A Simple Model
. Introduction In the last years, evolutionary-game theory has provided robust solutions to the problem of selection among multiple Nash equilibria in symmetric coordination games (Samuelson, 997). More
More informationPietro Dindo and Filippo Massari. The Wisdom of the Crowd in Dynamic Economies
Pietro Dindo and Filippo Massari The Wisdom of the Crowd in Dynamic Economies ISSN: 827-3580 No. 7/WP/207 Working Papers Department of Economics Ca Foscari University of Venice No. 7/WP/20 7 ISSN 827-3580
More informationLearning Methods for Online Prediction Problems. Peter Bartlett Statistics and EECS UC Berkeley
Learning Methods for Online Prediction Problems Peter Bartlett Statistics and EECS UC Berkeley Course Synopsis A finite comparison class: A = {1,..., m}. Converting online to batch. Online convex optimization.
More informationIntroduction We have considered many models with autonomous individuals or agents (eg. birth/death, citizens of cities, ants). Agents have been modell
Modelling Intentionality: The Gambler Rik Blok December 1, 1998 Introduction We have considered many models with autonomous individuals or agents (eg. birth/death, citizens of cities, ants). Agents have
More informationArea I: Contract Theory Question (Econ 206)
Theory Field Exam Summer 2011 Instructions You must complete two of the four areas (the areas being (I) contract theory, (II) game theory A, (III) game theory B, and (IV) psychology & economics). Be sure
More informationThe Heterogeneous Expectations Hypothesis: Some Evidence from the Lab
The : Some Evidence from the Lab Cars Hommes CeNDEF, Amsterdam School of Economics University of Amsterdam Evolution and Market Beavior in Economics and Finance Scuola Superiore Sant Anna, Pisa, Italia,
More informationA Necessary and Sufficient Condition for Convergence of Statistical to Strategic Equilibria of Market Games
A Necessary and Sufficient Condition for Convergence of Statistical to Strategic Equilibria of Market Games Dimitrios P. Tsomocos Dimitris Voliotis Abstract In our model, we treat a market game where traders
More informationGeneral Examination in Macroeconomic Theory SPRING 2013
HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS General Examination in Macroeconomic Theory SPRING 203 You have FOUR hours. Answer all questions Part A (Prof. Laibson): 48 minutes Part B (Prof. Aghion): 48
More informationErgodicity and Non-Ergodicity in Economics
Abstract An stochastic system is called ergodic if it tends in probability to a limiting form that is independent of the initial conditions. Breakdown of ergodicity gives rise to path dependence. We illustrate
More informationUncertainty and Disagreement in Equilibrium Models
Uncertainty and Disagreement in Equilibrium Models Nabil I. Al-Najjar & Northwestern University Eran Shmaya Tel Aviv University RUD, Warwick, June 2014 Forthcoming: Journal of Political Economy Motivation
More informationElimination of Arbitrage States in Asymmetric Information Models
Elimination of Arbitrage States in Asymmetric Information Models Bernard CORNET, Lionel DE BOISDEFFRE Abstract In a financial economy with asymmetric information and incomplete markets, we study how agents,
More informationBirgit Rudloff Operations Research and Financial Engineering, Princeton University
TIME CONSISTENT RISK AVERSE DYNAMIC DECISION MODELS: AN ECONOMIC INTERPRETATION Birgit Rudloff Operations Research and Financial Engineering, Princeton University brudloff@princeton.edu Alexandre Street
More informationEconomics of disagreement. financial intuition for the Rényi divergence
arxiv:1811.08308v1 [q-fin.gn] 20 Nov 2018 Economics of disagreement financial intuition for the Rényi divergence Andrei N. Soklakov Lack of accurate intuition is often cited as a scientific challenge,
More informationAmbiguity and the Centipede Game
Ambiguity and the Centipede Game Jürgen Eichberger, Simon Grant and David Kelsey Heidelberg University, Australian National University, University of Exeter. University of Exeter. June 2018 David Kelsey
More informationPatience and Ultimatum in Bargaining
Patience and Ultimatum in Bargaining Björn Segendorff Department of Economics Stockholm School of Economics PO Box 6501 SE-113 83STOCKHOLM SWEDEN SSE/EFI Working Paper Series in Economics and Finance No
More information1 Uncertainty. These notes correspond to chapter 2 of Jehle and Reny.
These notes correspond to chapter of Jehle and Reny. Uncertainty Until now we have considered our consumer s making decisions in a world with perfect certainty. However, we can extend the consumer theory
More informationInformation Choice in Macroeconomics and Finance.
Information Choice in Macroeconomics and Finance. Laura Veldkamp New York University, Stern School of Business, CEPR and NBER Spring 2009 1 Veldkamp What information consumes is rather obvious: It consumes
More informationEntry under an Information-Gathering Monopoly Alex Barrachina* June Abstract
Entry under an Information-Gathering onopoly Alex Barrachina* June 2016 Abstract The effects of information-gathering activities on a basic entry model with asymmetric information are analyzed. In the
More informationUncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6
1 Uncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6 1 A Two-Period Example Suppose the economy lasts only two periods, t =0, 1. The uncertainty arises in the income (wage) of period 1. Not that
More informationData Abundance and Asset Price Informativeness. On-Line Appendix
Data Abundance and Asset Price Informativeness On-Line Appendix Jérôme Dugast Thierry Foucault August 30, 07 This note is the on-line appendix for Data Abundance and Asset Price Informativeness. It contains
More informationCompetitive Equilibria in a Comonotone Market
Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang http://sas.uwaterloo.ca/ wang Department of Statistics and Actuarial Science University of Waterloo
More informationRSMG Working Paper Series. TITLE: The value of information and the value of awareness. Author: John Quiggin. Working Paper: R13_2
2013 TITLE: The value of information and the value of awareness 2011 RSMG Working Paper Series Risk and Uncertainty Program Author: John Quiggin Working Paper: R13_2 Schools of Economics and Political
More informationPsychology and Economics (Lecture 3)
Psychology and Economics (Lecture 3) Xavier Gabaix February 10, 2003 1 Discussion of traditional objections In the real world people would get it right Answer: there are many non-repeated interactions,
More informationTheory Field Examination Game Theory (209A) Jan Question 1 (duopoly games with imperfect information)
Theory Field Examination Game Theory (209A) Jan 200 Good luck!!! Question (duopoly games with imperfect information) Consider a duopoly game in which the inverse demand function is linear where it is positive
More informationPolitical Economy of Institutions and Development: Problem Set 1. Due Date: Thursday, February 23, in class.
Political Economy of Institutions and Development: 14.773 Problem Set 1 Due Date: Thursday, February 23, in class. Answer Questions 1-3. handed in. The other two questions are for practice and are not
More informationOptimal Insurance of Search Risk
Optimal Insurance of Search Risk Mikhail Golosov Yale University and NBER Pricila Maziero University of Pennsylvania Guido Menzio University of Pennsylvania and NBER November 2011 Introduction Search and
More informationSentiments and Aggregate Fluctuations
Sentiments and Aggregate Fluctuations Jess Benhabib Pengfei Wang Yi Wen October 15, 2013 Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 1 / 43 Introduction
More informationThis corresponds to a within-subject experiment: see same subject make choices from different menus.
Testing Revealed Preference Theory, I: Methodology The revealed preference theory developed last time applied to a single agent. This corresponds to a within-subject experiment: see same subject make choices
More informationThis is designed for one 75-minute lecture using Games and Information. October 3, 2006
This is designed for one 75-minute lecture using Games and Information. October 3, 2006 1 7 Moral Hazard: Hidden Actions PRINCIPAL-AGENT MODELS The principal (or uninformed player) is the player who has
More informationEcon 101A Problem Set 6 Solutions Due on Monday Dec. 9. No late Problem Sets accepted, sorry!
Econ 0A Problem Set 6 Solutions Due on Monday Dec. 9. No late Problem Sets accepted, sry! This Problem set tests the knowledge that you accumulated mainly in lectures 2 to 26. The problem set is focused
More informationECON4510 Finance Theory Lecture 2
ECON4510 Finance Theory Lecture 2 Diderik Lund Department of Economics University of Oslo 26 August 2013 Diderik Lund, Dept. of Economics, UiO ECON4510 Lecture 2 26 August 2013 1 / 31 Risk aversion and
More informationGambling and Information Theory
Gambling and Information Theory Giulio Bertoli UNIVERSITEIT VAN AMSTERDAM December 17, 2014 Overview Introduction Kelly Gambling Horse Races and Mutual Information Some Facts Shannon (1948): definitions/concepts
More informationSome Remarks on Alternating Temporal Epistemic Logic
Some Remarks on Alternating Temporal Epistemic Logic Corrected version: July 2003 Wojciech Jamroga Parlevink Group, University of Twente, Netherlands Institute of Mathematics, University of Gdansk, Poland
More informationSpeculation and the Bond Market: An Empirical No-arbitrage Framework
Online Appendix to the paper Speculation and the Bond Market: An Empirical No-arbitrage Framework October 5, 2015 Part I: Maturity specific shocks in affine and equilibrium models This Appendix present
More informationSelecting Efficient Correlated Equilibria Through Distributed Learning. Jason R. Marden
1 Selecting Efficient Correlated Equilibria Through Distributed Learning Jason R. Marden Abstract A learning rule is completely uncoupled if each player s behavior is conditioned only on his own realized
More informationSpeculative Investor Behavior and Learning
Speculative Investor Behavior and Learning QJE 1996 Stephen Morris presentation by Jonathan R. L. Halket Speculative Investor Behavior and Learning by Stephen Morris p. 1/13 Introduction and Motivation
More informationNotes on Recursive Utility. Consider the setting of consumption in infinite time under uncertainty as in
Notes on Recursive Utility Consider the setting of consumption in infinite time under uncertainty as in Section 1 (or Chapter 29, LeRoy & Werner, 2nd Ed.) Let u st be the continuation utility at s t. That
More informationEndogenous Information Choice
Endogenous Information Choice Lecture 7 February 11, 2015 An optimizing trader will process those prices of most importance to his decision problem most frequently and carefully, those of less importance
More informationIntroduction to Game Theory
Introduction to Game Theory Part 3. Dynamic games of incomplete information Chapter 3. Job Market Signaling Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe Martins-da-Rocha
More information1 The Basic RBC Model
IHS 2016, Macroeconomics III Michael Reiter Ch. 1: Notes on RBC Model 1 1 The Basic RBC Model 1.1 Description of Model Variables y z k L c I w r output level of technology (exogenous) capital at end of
More informationSecond Welfare Theorem
Second Welfare Theorem Econ 2100 Fall 2015 Lecture 18, November 2 Outline 1 Second Welfare Theorem From Last Class We want to state a prove a theorem that says that any Pareto optimal allocation is (part
More informationRobert Wilson (1977), A Bidding Model of Perfect Competition, Review of Economic
Econ 805 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 10 Oct 7 2008 No lecture on Thursday New HW, and solutions to last one, online Today s papers: Robert Wilson (1977), A Bidding Model of Perfect
More informationDeceptive Advertising with Rational Buyers
Deceptive Advertising with Rational Buyers September 6, 016 ONLINE APPENDIX In this Appendix we present in full additional results and extensions which are only mentioned in the paper. In the exposition
More informationDepartment of Economics The Ohio State University Final Exam Questions and Answers Econ 8712
Prof. Peck Fall 20 Department of Economics The Ohio State University Final Exam Questions and Answers Econ 872. (0 points) The following economy has two consumers, two firms, and three goods. Good is leisure/labor.
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Adverse Selection We have now completed our basic analysis of the adverse selection model This model has been applied and extended in literally thousands of ways
More information1 Gambler s Ruin Problem
1 Gambler s Ruin Problem Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble either wins $1 or loses $1 independent of the past with probabilities p and q = 1
More informationAdvanced Microeconomics
Advanced Microeconomics Partial and General Equilibrium Giorgio Fagiolo giorgio.fagiolo@sssup.it http://www.lem.sssup.it/fagiolo/welcome.html LEM, Sant Anna School of Advanced Studies, Pisa (Italy) Part
More informationAmbiguity and Information Processing in a Model of Intermediary Asset Pricing
Ambiguity and Information Processing in a Model of Intermediary Asset Pricing Leyla Jianyu Han 1 Kenneth Kasa 2 Yulei Luo 1 1 The University of Hong Kong 2 Simon Fraser University December 15, 218 1 /
More informationA New Interpretation of Information Rate
A New Interpretation of Information Rate By J. L. KELLY. JR. (Manuscript received March 2, 956) If the input symbols to a communication channel represent the outcomes of a chance event on which bets are
More informationECOM 009 Macroeconomics B. Lecture 2
ECOM 009 Macroeconomics B Lecture 2 Giulio Fella c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 2 40/197 Aim of consumption theory Consumption theory aims at explaining consumption/saving decisions
More informationThe Real Business Cycle Model
The Real Business Cycle Model Macroeconomics II 2 The real business cycle model. Introduction This model explains the comovements in the fluctuations of aggregate economic variables around their trend.
More informationNoncooperative Games, Couplings Constraints, and Partial Effi ciency
Noncooperative Games, Couplings Constraints, and Partial Effi ciency Sjur Didrik Flåm University of Bergen, Norway Background Customary Nash equilibrium has no coupling constraints. Here: coupling constraints
More informationMULTI-BELIEF RATIONAL-EXPECTATIONS EQUILIBRIA: INDETERMINACY, COMPLEXITY AND SUSTAINED DEFLATION
JSPS Grants-in-Aid for Scientific Research (S) Understanding Persistent Deflation in Japan Working Paper Series No. 058 December 2014 MULTI-BELIEF RATIONAL-EXPECTATIONS EQUILIBRIA: INDETERMINACY, COMPLEXITY
More informationMeasuring the informativeness of economic actions and. market prices 1. Philip Bond, University of Washington. September 2014
Measuring the informativeness of economic actions and market prices 1 Philip Bond, University of Washington September 2014 1 I thank Raj Singh for some very constructive conversations, along with a seminar
More informationBargaining, Contracts, and Theories of the Firm. Dr. Margaret Meyer Nuffield College
Bargaining, Contracts, and Theories of the Firm Dr. Margaret Meyer Nuffield College 2015 Course Overview 1. Bargaining 2. Hidden information and self-selection Optimal contracting with hidden information
More informationThe ambiguous impact of contracts on competition in the electricity market Yves Smeers
The ambiguous impact of contracts on competition in the electricity market Yves Smeers joint work with Frederic Murphy Climate Policy and Long Term Decisions-Investment and R&D, Bocconi University, Milan,
More informationBanks, depositors and liquidity shocks: long term vs short term interest rates in a model of adverse selection
Banks, depositors and liquidity shocks: long term vs short term interest rates in a model of adverse selection Geethanjali Selvaretnam Abstract This model takes into consideration the fact that depositors
More informationFINM6900 Finance Theory Noisy Rational Expectations Equilibrium for Multiple Risky Assets
FINM69 Finance Theory Noisy Rational Expectations Equilibrium for Multiple Risky Assets February 3, 212 Reference Anat R. Admati, A Noisy Rational Expectations Equilibrium for Multi-Asset Securities Markets,
More informationPersuading Skeptics and Reaffirming Believers
Persuading Skeptics and Reaffirming Believers May, 31 st, 2014 Becker-Friedman Institute Ricardo Alonso and Odilon Camara Marshall School of Business - USC Introduction Sender wants to influence decisions
More informationECO 317 Economics of Uncertainty. Lectures: Tu-Th , Avinash Dixit. Precept: Fri , Andrei Rachkov. All in Fisher Hall B-01
ECO 317 Economics of Uncertainty Lectures: Tu-Th 3.00-4.20, Avinash Dixit Precept: Fri 10.00-10.50, Andrei Rachkov All in Fisher Hall B-01 1 No.1 Thu. Sep. 17 ECO 317 Fall 09 RISK MARKETS Intrade: http://www.intrade.com
More informationIn the Ramsey model we maximized the utility U = u[c(t)]e nt e t dt. Now
PERMANENT INCOME AND OPTIMAL CONSUMPTION On the previous notes we saw how permanent income hypothesis can solve the Consumption Puzzle. Now we use this hypothesis, together with assumption of rational
More informationAdvanced Microeconomics
Advanced Microeconomics Leonardo Felli EC441: Room D.106, Z.332, D.109 Lecture 8 bis: 24 November 2004 Monopoly Consider now the pricing behavior of a profit maximizing monopolist: a firm that is the only
More informationA Study on Lucas' ``Expectations and the Neutrality of Money. Masayuki Otaki (Institute of Social Science, University of Tokyo)
DBJ Discussion Paper Series, No.03 A Study on Lucas' ``Epectations and the Neutrality of Money Masayuki Otaki (Institute of Social Science, University of Tokyo) July 0 Discussion Papers are a series of
More informationUse of Eigen values and eigen vectors to calculate higher transition probabilities
The Lecture Contains : Markov-Bernoulli Chain Note Assignments Random Walks which are correlated Actual examples of Markov Chains Examples Use of Eigen values and eigen vectors to calculate higher transition
More informationCournot and Bertrand Competition in a Differentiated Duopoly with Endogenous Technology Adoption *
ANNALS OF ECONOMICS AND FINANCE 16-1, 231 253 (2015) Cournot and Bertrand Competition in a Differentiated Duopoly with Endogenous Technology Adoption * Hongkun Ma School of Economics, Shandong University,
More informationChoice under uncertainty
Choice under uncertainty Expected utility theory The agent chooses among a set of risky alternatives (lotteries) Description of risky alternatives (lotteries) a lottery L = a random variable on a set of
More informationEconomics 2010c: Lectures 9-10 Bellman Equation in Continuous Time
Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time David Laibson 9/30/2014 Outline Lectures 9-10: 9.1 Continuous-time Bellman Equation 9.2 Application: Merton s Problem 9.3 Application:
More informationMotivation Non-linear Rational Expectations The Permanent Income Hypothesis The Log of Gravity Non-linear IV Estimation Summary.
Econometrics I Department of Economics Universidad Carlos III de Madrid Master in Industrial Economics and Markets Outline Motivation 1 Motivation 2 3 4 5 Motivation Hansen's contributions GMM was developed
More informationMULTIPLE-BELIEF RATIONAL-EXPECTATIONS EQUILIBRIA IN OLG MODELS WITH AMBIGUITY
MULTIPLE-BELIEF RATIONAL-EXPECTATIONS EQUILIBRIA IN OLG MODELS WITH AMBIGUITY by Kiyohiko G. Nishimura Faculty of Economics The University of Tokyo and Hiroyuki Ozaki Faculty of Economics Tohoku University
More informationThe Non-Existence of Representative Agents
The Non-Existence of Representative Agents Matthew O. Jackson and Leeat Yariv November 2015 Abstract We characterize environments in which there exists a representative agent: an agent who inherits the
More informationComprehensive Exam. Macro Spring 2014 Retake. August 22, 2014
Comprehensive Exam Macro Spring 2014 Retake August 22, 2014 You have a total of 180 minutes to complete the exam. If a question seems ambiguous, state why, sharpen it up and answer the sharpened-up question.
More informationAre Probabilities Used in Markets? 1
Journal of Economic Theory 91, 8690 (2000) doi:10.1006jeth.1999.2590, available online at http:www.idealibrary.com on NOTES, COMMENTS, AND LETTERS TO THE EDITOR Are Probabilities Used in Markets? 1 Larry
More informationHomework #6 (10/18/2017)
Homework #6 (0/8/207). Let G be the set of compound gambles over a finite set of deterministic payoffs {a, a 2,...a n } R +. A decision maker s preference relation over compound gambles can be represented
More information2008/73. On the Golden Rule of Capital Accumulation Under Endogenous Longevity. David DE LA CROIX Grégory PONTHIERE
2008/73 On the Golden Rule of Capital Accumulation Under Endogenous Longevity David DE LA CROIX Grégory PONTHIERE On the Golden Rule of Capital Accumulation under Endogenous Longevity David de la Croix
More informationRisk Aversion, Beliefs, and Prediction Market Equilibrium
Risk version, Beliefs, and Prediction Market Equilibrium Steven Gjerstad Economic Science Laboratory McClelland Hall 116 University of rizona Tucson, rizona 85721 gjerstad@econlab.arizona.edu January 2005
More informationOn the Unique D1 Equilibrium in the Stackelberg Model with Asymmetric Information Janssen, M.C.W.; Maasland, E.
Tilburg University On the Unique D1 Equilibrium in the Stackelberg Model with Asymmetric Information Janssen, M.C.W.; Maasland, E. Publication date: 1997 Link to publication General rights Copyright and
More informationKlaus Kultti Hannu Salonen Demented Prisoners. Aboa Centre for Economics
Klaus Kultti Hannu Salonen Demented Prisoners Aboa Centre for Economics Discussion Paper No. 43 Turku 2009 Copyright Author(s) ISSN 1796-3133 Printed in Uniprint Turku 2009 Klaus Kultti Hannu Salonen Demented
More informationWorking Paper EQUILIBRIUM SELECTION IN COMMON-VALUE SECOND-PRICE AUCTIONS. Heng Liu
Working Paper EQUILIBRIUM SELECTION IN COMMON-VALUE SECOND-PRICE AUCTIONS Heng Liu This note considers equilibrium selection in common-value secondprice auctions with two bidders. We show that for each
More informationIntroduction to Game Theory
Introduction to Game Theory Part 2. Dynamic games of complete information Chapter 2. Two-stage games of complete but imperfect information Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas
More informationWars of Attrition with Budget Constraints
Wars of Attrition with Budget Constraints Gagan Ghosh Bingchao Huangfu Heng Liu October 19, 2017 (PRELIMINARY AND INCOMPLETE: COMMENTS WELCOME) Abstract We study wars of attrition between two bidders who
More informationCoherence with Proper Scoring Rules
Coherence with Proper Scoring Rules Mark Schervish, Teddy Seidenfeld, and Joseph (Jay) Kadane Mark Schervish Joseph ( Jay ) Kadane Coherence with Proper Scoring Rules ILC, Sun Yat-Sen University June 2010
More informationModels of Wage Dynamics
Models of Wage Dynamics Toshihiko Mukoyama Department of Economics Concordia University and CIREQ mukoyama@alcor.concordia.ca December 13, 2005 1 Introduction This paper introduces four different models
More informationLearning from Others Outcomes
Learning from Others Outcomes Alexander Wolitzky MIT BFI, July 21, 2017 Wolitzky (MIT) Learning from Others Outcomes BFI, July 21, 2017 1 / 53 Introduction Scarcity of Cost-Saving Innovation Development
More informationTHEORIES ON AUCTIONS WITH PARTICIPATION COSTS. A Dissertation XIAOYONG CAO
THEORIES ON AUCTIONS WITH PARTICIPATION COSTS A Dissertation by XIAOYONG CAO Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree
More informationRecitation 7: Uncertainty. Xincheng Qiu
Econ 701A Fall 2018 University of Pennsylvania Recitation 7: Uncertainty Xincheng Qiu (qiux@sas.upenn.edu 1 Expected Utility Remark 1. Primitives: in the basic consumer theory, a preference relation is
More informationComplex Systems Workshop Lecture III: Behavioral Asset Pricing Model with Heterogeneous Beliefs
Complex Systems Workshop Lecture III: Behavioral Asset Pricing Model with Heterogeneous Beliefs Cars Hommes CeNDEF, UvA CEF 2013, July 9, Vancouver Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver
More informationWavelet based sample entropy analysis: A new method to test weak form market efficiency
Theoretical and Applied Economics Volume XXI (2014), No. 8(597), pp. 19-26 Fet al Wavelet based sample entropy analysis: A new method to test weak form market efficiency Anoop S. KUMAR University of Hyderabad,
More informationECON 5118 Macroeconomic Theory
ECON 5118 Macroeconomic Theory Winter 013 Test 1 February 1, 013 Answer ALL Questions Time Allowed: 1 hour 0 min Attention: Please write your answers on the answer book provided Use the right-side pages
More informationPreferences for Randomization and Anticipation
Preferences for Randomization and Anticipation Yosuke Hashidate Abstract In decision theory, it is not generally postulated that decision makers randomize their choices. In contrast, in real life, even
More informationNBER WORKING PAPER SERIES MARKET SELECTION. Leonid Kogan Stephen Ross Jiang Wang Mark M. Westerfield
NBER WORKING PAPER SERIES MARKET SELECTION Leonid Kogan Stephen Ross Jiang Wang Mark M. Westerfield Working Paper 15189 http://www.nber.org/papers/w15189 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts
More informationECON4515 Finance theory 1 Diderik Lund, 5 May Perold: The CAPM
Perold: The CAPM Perold starts with a historical background, the development of portfolio theory and the CAPM. Points out that until 1950 there was no theory to describe the equilibrium determination of
More informationDISCUSSION PAPER SERIES
DISCUSSION PAPER SERIES IN ECONOMICS AND MANAGEMENT Strategic Incentives for Managers in Contests Matthias Kräkel Discussion Paper No. 01-08 GERMAN ECONOMIC ASSOCIATION OF BUSINESS ADMINISTRATION - GEABA
More informationIdentification and Estimation of Bidders Risk Aversion in. First-Price Auctions
Identification and Estimation of Bidders Risk Aversion in First-Price Auctions Isabelle Perrigne Pennsylvania State University Department of Economics University Park, PA 16802 Phone: (814) 863-2157, Fax:
More informationLecture 3: Huggett s 1993 model. 1 Solving the savings problem for an individual consumer
UNIVERSITY OF WESTERN ONTARIO LONDON ONTARIO Paul Klein Office: SSC 4044 Extension: 85484 Email: pklein2@uwo.ca URL: http://paulklein.ca/newsite/teaching/619.php Economics 9619 Computational methods in
More informationA Generic Bound on Cycles in Two-Player Games
A Generic Bound on Cycles in Two-Player Games David S. Ahn February 006 Abstract We provide a bound on the size of simultaneous best response cycles for generic finite two-player games. The bound shows
More information